SPECTRAL ASYMPTOTICS WITH A REMAINDER ESTIMATE OF THE NEUMANN LAPLACIAN ON HORNS - THE CASE OF THE RAPIDLY GROWING COUNTING FUNCTION

We study the Neumann Laplacian -Delta(N)(Omega): in unbounded regions of the form Omega = {(t, x)/t > 0, f(t)(-1) x is an element of Omega') , where Omega' subset of Rn-1 is a bounded open set with the Lipschitz boundary and f decays in such a way that the spectrum of -Delta(N)(Om ega): is discrete but the counting function N(lambda, -Delta(N)(Omega) ) of the spectrum grows faster than a power of lambda, a typical examp le being f(t) = exp (-t 1n...1n t), for t greater than or equal to t(0 ). We compute the principal term of the asymptotics of N(lambda, -Delt a(N)(Omega)), with a remainder estimate.